Exercise 11, § 3

With the hypotheses and notation of Exercise 26 of § 2, suppose that each of the groups $G_i$ is Hausdorff and has a completion. Show that the local product G of the $G_i$, relative to a family $(K_i)$ of open normal subgroups, has a completion $\hat{G}$ which is isomorphic to the local product of the $\hat{G}_i$, relative to the closures $\overline{K}_i$ of the $K_i$.